41 research outputs found
Tippe Top Inversion as a Dissipation-Induced Instability
By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell--Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597--609]. A linear analysis of the equations of motion reveals that the only equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell--Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top
Dissipation-Induced Heteroclinic Orbits in Tippe Tops
This paper demonstrates that the conditions for the existence of a dissipation-induced heteroclinic orbit between the inverted and noninverted states of a tippe top are determined by a complex version of the equations for a simple harmonic oscillator: the modified Maxwell–Bloch equations. A standard linear analysis reveals that the modified Maxwell–Bloch equations describe the spectral instability of the noninverted state and Lyapunov stability of the inverted state. Standard nonlinear analysis based on the energy momentum method gives necessary and sufficient conditions for the existence of a dissipation-induced connecting orbit between these relative equilibria
Motion of the Tippe Top : Gyroscopic Balance Condition and Stability
We reexamine a very classical problem, the spinning behavior of the tippe top
on a horizontal table. The analysis is made for an eccentric sphere version of
the tippe top, assuming a modified Coulomb law for the sliding friction, which
is a continuous function of the slip velocity at the point of
contact and vanishes at . We study the relevance of the gyroscopic
balance condition (GBC), which was discovered to hold for a rapidly spinning
hard-boiled egg by Moffatt and Shimomura, to the inversion phenomenon of the
tippe top. We introduce a variable so that corresponds to the GBC
and analyze the behavior of . Contrary to the case of the spinning egg,
the GBC for the tippe top is not fulfilled initially. But we find from
simulation that for those tippe tops which will turn over, the GBC will soon be
satisfied approximately. It is shown that the GBC and the geometry lead to the
classification of tippe tops into three groups: The tippe tops of Group I never
flip over however large a spin they are given. Those of Group II show a
complete inversion and the tippe tops of Group III tend to turn over up to a
certain inclination angle such that , when they are
spun sufficiently rapidly. There exist three steady states for the spinning
motion of the tippe top. Giving a new criterion for stability, we examine the
stability of these states in terms of the initial spin velocity . And we
obtain a critical value of the initial spin which is required for the
tippe top of Group II to flip over up to the completely inverted position.Comment: 52 pages, 11 figures, to be published in SIAM Journal on Applied
Dynamical Syste
Friction, Free Axes of Rotation and Entropy
Friction forces acting on rotators may promote their alignment and therefore
eliminate degrees of freedom in their movement. The alignment of rotators by
friction force was demonstrated by experiments performed with different
spinners, demonstrating how friction generates negentropy in a system of
rotators. A gas of rigid rotators influenced by friction force is considered.
The orientational negentropy generated by a friction force was estimated with
the Sackur-Tetrode equation. The minimal change in total entropy of a system of
rotators, corresponding to their eventual alignment, decreases with
temperature. The reported effect may be of the primary importance for a phase
equilibrium and motion of ubiquitous colloidal and granular systems.Comment: 10 pages, 7 image
Dynamics of the Tippe Top -- properties of numerical solutions versus the dynamical equations
We study the relationship between numerical solutions for inverting Tippe Top
and the structure of the dynamical equations. The numerical solutions confirm
oscillatory behaviour of the inclination angle for the symmetry
axis of the Tippe Top. They also reveal further fine features of the dynamics
of inverting solutions defining the time of inversion. These features are
partially understood on the basis of the underlying dynamical equations
Dynamics of the Tippe Top via Routhian Reduction
We consider a tippe top modeled as an eccentric sphere, spinning on a
horizontal table and subject to a sliding friction. Ignoring translational
effects, we show that the system is reducible using a Routhian reduction
technique. The reduced system is a two dimensional system of second order
differential equations, that allows an elegant and compact way to retrieve the
classification of tippe tops in six groups as proposed in [1] according to the
existence and stability type of the steady states.Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten
term in de linearized system is adde
Tippe Top Equations and Equations for the Related Mechanical Systems
The equations of motion for the rolling and gliding Tippe Top (TT) are
nonintegrable and difficult to analyze. The only existing arguments about TT
inversion are based on analysis of stability of asymptotic solutions and the
LaSalle type theorem. They do not explain the dynamics of inversion. To
approach this problem we review and analyze here the equations of motion for
the rolling and gliding TT in three equivalent forms, each one providing
different bits of information about motion of TT. They lead to the main
equation for the TT, which describes well the oscillatory character of motion
of the symmetry axis during the inversion. We show also that
the equations of motion of TT give rise to equations of motion for two other
simpler mechanical systems: the gliding heavy symmetric top and the gliding
eccentric cylinder. These systems can be of aid in understanding the dynamics
of the inverting TT
Towards a prototype of a spherical tippe top
Among spinning objects, the tippe top exhibits one of the most bizarre and counterintuitive behaviours. The commercially available tippe tops basically consist of a section of a sphere with a rod. After spinning on its rounded body, the top flips over and continues spinning on the stem. It is the friction with the bottom surface and the position of the center of mass below the centre of curvature that cause the tippe top to rise its centre of mass while continuing rotating around its symmetry axis (through the stem). The commonly used simplified mathematical model for the tippe top is a sphere whose mass distribution is axially but not spherically symmetric, spinning on a flat surface subject to a small friction force that is due to sliding.
Adopting a bifurcation theory point of view we reach a global geometric understanding of the phase diagram of this dynamical system. According to the eccentricity of the sphere and the Jellet invariant (which includes information on the initial angular velocity) three main different dynamical behaviours are distinguished: tipping, non-tipping, hanging (i.e. the top rises but converges to an intermediate state instead of rising all the way to the vertical state). Subclasses according to the stability of relative equilibria can further be distinguished. Since our concern is the degree of confidence in the mathematical model predictions, we applied 3D-printing and rapid prototyping to manufacture a ’3-in-1 toy’ that could catch the three main characteristics defining the three main groups in the classification of spherical tippe tops as mentioned above. This ’toy’ is suitable to validate the mathematical model qualitatively and quantitatively